Properties

Label 135.4.a.f.1.1
Level $135$
Weight $4$
Character 135.1
Self dual yes
Analytic conductor $7.965$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 135 = 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 135.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(7.96525785077\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.5637.1
Defining polynomial: \( x^{3} - x^{2} - 23x + 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(5.20067\) of defining polynomial
Character \(\chi\) \(=\) 135.1

$q$-expansion

\(f(q)\) \(=\) \(q-5.20067 q^{2} +19.0470 q^{4} +5.00000 q^{5} +24.4013 q^{7} -57.4517 q^{8} +O(q^{10})\) \(q-5.20067 q^{2} +19.0470 q^{4} +5.00000 q^{5} +24.4013 q^{7} -57.4517 q^{8} -26.0034 q^{10} +28.9839 q^{11} -65.3919 q^{13} -126.903 q^{14} +146.411 q^{16} -68.1718 q^{17} +104.424 q^{19} +95.2349 q^{20} -150.736 q^{22} +154.807 q^{23} +25.0000 q^{25} +340.082 q^{26} +464.772 q^{28} +205.658 q^{29} -18.2497 q^{31} -301.824 q^{32} +354.539 q^{34} +122.007 q^{35} -337.613 q^{37} -543.076 q^{38} -287.258 q^{40} +195.969 q^{41} +334.882 q^{43} +552.055 q^{44} -805.098 q^{46} +5.00398 q^{47} +252.425 q^{49} -130.017 q^{50} -1245.52 q^{52} +319.965 q^{53} +144.919 q^{55} -1401.90 q^{56} -1069.56 q^{58} -430.611 q^{59} +594.581 q^{61} +94.9106 q^{62} +398.396 q^{64} -326.960 q^{65} +195.876 q^{67} -1298.47 q^{68} -634.517 q^{70} -425.955 q^{71} +929.193 q^{73} +1755.82 q^{74} +1988.96 q^{76} +707.245 q^{77} +24.4296 q^{79} +732.057 q^{80} -1019.17 q^{82} +545.859 q^{83} -340.859 q^{85} -1741.61 q^{86} -1665.17 q^{88} +84.1332 q^{89} -1595.65 q^{91} +2948.60 q^{92} -26.0241 q^{94} +522.121 q^{95} +827.613 q^{97} -1312.78 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{2} + 23 q^{4} + 15 q^{5} + 44 q^{7} - 36 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - q^{2} + 23 q^{4} + 15 q^{5} + 44 q^{7} - 36 q^{8} - 5 q^{10} + 38 q^{11} + 28 q^{13} - 108 q^{14} + 191 q^{16} - 19 q^{17} + 187 q^{19} + 115 q^{20} + 122 q^{22} - 81 q^{23} + 75 q^{25} + 416 q^{26} + 410 q^{28} + 160 q^{29} + 227 q^{31} - 569 q^{32} + 17 q^{34} + 220 q^{35} + 78 q^{37} - 757 q^{38} - 180 q^{40} - 338 q^{41} + 22 q^{43} + 1636 q^{44} - 1425 q^{46} - 472 q^{47} - 197 q^{49} - 25 q^{50} - 1566 q^{52} + 521 q^{53} + 190 q^{55} - 1254 q^{56} - 2096 q^{58} + 140 q^{59} + 595 q^{61} + 1407 q^{62} - 918 q^{64} + 140 q^{65} + 878 q^{67} - 3053 q^{68} - 540 q^{70} - 602 q^{71} + 1294 q^{73} + 2878 q^{74} + 525 q^{76} + 288 q^{77} + 629 q^{79} + 955 q^{80} - 1682 q^{82} - 1287 q^{83} - 95 q^{85} - 3730 q^{86} - 858 q^{88} + 2154 q^{89} - 440 q^{91} + 1959 q^{92} - 1108 q^{94} + 935 q^{95} + 1392 q^{97} - 2693 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −5.20067 −1.83871 −0.919357 0.393424i \(-0.871291\pi\)
−0.919357 + 0.393424i \(0.871291\pi\)
\(3\) 0 0
\(4\) 19.0470 2.38087
\(5\) 5.00000 0.447214
\(6\) 0 0
\(7\) 24.4013 1.31755 0.658774 0.752341i \(-0.271075\pi\)
0.658774 + 0.752341i \(0.271075\pi\)
\(8\) −57.4517 −2.53903
\(9\) 0 0
\(10\) −26.0034 −0.822298
\(11\) 28.9839 0.794451 0.397226 0.917721i \(-0.369973\pi\)
0.397226 + 0.917721i \(0.369973\pi\)
\(12\) 0 0
\(13\) −65.3919 −1.39511 −0.697556 0.716530i \(-0.745729\pi\)
−0.697556 + 0.716530i \(0.745729\pi\)
\(14\) −126.903 −2.42260
\(15\) 0 0
\(16\) 146.411 2.28768
\(17\) −68.1718 −0.972593 −0.486296 0.873794i \(-0.661653\pi\)
−0.486296 + 0.873794i \(0.661653\pi\)
\(18\) 0 0
\(19\) 104.424 1.26087 0.630435 0.776242i \(-0.282876\pi\)
0.630435 + 0.776242i \(0.282876\pi\)
\(20\) 95.2349 1.06476
\(21\) 0 0
\(22\) −150.736 −1.46077
\(23\) 154.807 1.40345 0.701727 0.712446i \(-0.252413\pi\)
0.701727 + 0.712446i \(0.252413\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 340.082 2.56521
\(27\) 0 0
\(28\) 464.772 3.13691
\(29\) 205.658 1.31689 0.658443 0.752631i \(-0.271215\pi\)
0.658443 + 0.752631i \(0.271215\pi\)
\(30\) 0 0
\(31\) −18.2497 −0.105734 −0.0528668 0.998602i \(-0.516836\pi\)
−0.0528668 + 0.998602i \(0.516836\pi\)
\(32\) −301.824 −1.66736
\(33\) 0 0
\(34\) 354.539 1.78832
\(35\) 122.007 0.589226
\(36\) 0 0
\(37\) −337.613 −1.50009 −0.750044 0.661387i \(-0.769968\pi\)
−0.750044 + 0.661387i \(0.769968\pi\)
\(38\) −543.076 −2.31838
\(39\) 0 0
\(40\) −287.258 −1.13549
\(41\) 195.969 0.746469 0.373234 0.927737i \(-0.378249\pi\)
0.373234 + 0.927737i \(0.378249\pi\)
\(42\) 0 0
\(43\) 334.882 1.18765 0.593826 0.804594i \(-0.297617\pi\)
0.593826 + 0.804594i \(0.297617\pi\)
\(44\) 552.055 1.89149
\(45\) 0 0
\(46\) −805.098 −2.58055
\(47\) 5.00398 0.0155299 0.00776496 0.999970i \(-0.497528\pi\)
0.00776496 + 0.999970i \(0.497528\pi\)
\(48\) 0 0
\(49\) 252.425 0.735934
\(50\) −130.017 −0.367743
\(51\) 0 0
\(52\) −1245.52 −3.32158
\(53\) 319.965 0.829256 0.414628 0.909991i \(-0.363912\pi\)
0.414628 + 0.909991i \(0.363912\pi\)
\(54\) 0 0
\(55\) 144.919 0.355289
\(56\) −1401.90 −3.34529
\(57\) 0 0
\(58\) −1069.56 −2.42138
\(59\) −430.611 −0.950182 −0.475091 0.879937i \(-0.657585\pi\)
−0.475091 + 0.879937i \(0.657585\pi\)
\(60\) 0 0
\(61\) 594.581 1.24800 0.624002 0.781422i \(-0.285505\pi\)
0.624002 + 0.781422i \(0.285505\pi\)
\(62\) 94.9106 0.194414
\(63\) 0 0
\(64\) 398.396 0.778118
\(65\) −326.960 −0.623913
\(66\) 0 0
\(67\) 195.876 0.357166 0.178583 0.983925i \(-0.442849\pi\)
0.178583 + 0.983925i \(0.442849\pi\)
\(68\) −1298.47 −2.31562
\(69\) 0 0
\(70\) −634.517 −1.08342
\(71\) −425.955 −0.711994 −0.355997 0.934487i \(-0.615859\pi\)
−0.355997 + 0.934487i \(0.615859\pi\)
\(72\) 0 0
\(73\) 929.193 1.48978 0.744889 0.667188i \(-0.232502\pi\)
0.744889 + 0.667188i \(0.232502\pi\)
\(74\) 1755.82 2.75824
\(75\) 0 0
\(76\) 1988.96 3.00197
\(77\) 707.245 1.04673
\(78\) 0 0
\(79\) 24.4296 0.0347917 0.0173959 0.999849i \(-0.494462\pi\)
0.0173959 + 0.999849i \(0.494462\pi\)
\(80\) 732.057 1.02308
\(81\) 0 0
\(82\) −1019.17 −1.37254
\(83\) 545.859 0.721877 0.360938 0.932590i \(-0.382456\pi\)
0.360938 + 0.932590i \(0.382456\pi\)
\(84\) 0 0
\(85\) −340.859 −0.434957
\(86\) −1741.61 −2.18375
\(87\) 0 0
\(88\) −1665.17 −2.01714
\(89\) 84.1332 0.100203 0.0501017 0.998744i \(-0.484045\pi\)
0.0501017 + 0.998744i \(0.484045\pi\)
\(90\) 0 0
\(91\) −1595.65 −1.83813
\(92\) 2948.60 3.34144
\(93\) 0 0
\(94\) −26.0241 −0.0285551
\(95\) 522.121 0.563879
\(96\) 0 0
\(97\) 827.613 0.866303 0.433152 0.901321i \(-0.357401\pi\)
0.433152 + 0.901321i \(0.357401\pi\)
\(98\) −1312.78 −1.35317
\(99\) 0 0
\(100\) 476.174 0.476174
\(101\) −823.576 −0.811375 −0.405688 0.914012i \(-0.632968\pi\)
−0.405688 + 0.914012i \(0.632968\pi\)
\(102\) 0 0
\(103\) 1171.19 1.12040 0.560198 0.828359i \(-0.310725\pi\)
0.560198 + 0.828359i \(0.310725\pi\)
\(104\) 3756.87 3.54223
\(105\) 0 0
\(106\) −1664.03 −1.52477
\(107\) −1023.21 −0.924460 −0.462230 0.886760i \(-0.652951\pi\)
−0.462230 + 0.886760i \(0.652951\pi\)
\(108\) 0 0
\(109\) −403.647 −0.354700 −0.177350 0.984148i \(-0.556753\pi\)
−0.177350 + 0.984148i \(0.556753\pi\)
\(110\) −753.678 −0.653276
\(111\) 0 0
\(112\) 3572.63 3.01413
\(113\) −1082.20 −0.900931 −0.450465 0.892794i \(-0.648742\pi\)
−0.450465 + 0.892794i \(0.648742\pi\)
\(114\) 0 0
\(115\) 774.033 0.627643
\(116\) 3917.16 3.13534
\(117\) 0 0
\(118\) 2239.46 1.74711
\(119\) −1663.48 −1.28144
\(120\) 0 0
\(121\) −490.935 −0.368847
\(122\) −3092.22 −2.29473
\(123\) 0 0
\(124\) −347.601 −0.251738
\(125\) 125.000 0.0894427
\(126\) 0 0
\(127\) −774.132 −0.540890 −0.270445 0.962735i \(-0.587171\pi\)
−0.270445 + 0.962735i \(0.587171\pi\)
\(128\) 342.664 0.236621
\(129\) 0 0
\(130\) 1700.41 1.14720
\(131\) 1214.04 0.809702 0.404851 0.914383i \(-0.367323\pi\)
0.404851 + 0.914383i \(0.367323\pi\)
\(132\) 0 0
\(133\) 2548.09 1.66126
\(134\) −1018.69 −0.656726
\(135\) 0 0
\(136\) 3916.58 2.46944
\(137\) −2300.15 −1.43441 −0.717207 0.696860i \(-0.754580\pi\)
−0.717207 + 0.696860i \(0.754580\pi\)
\(138\) 0 0
\(139\) −1355.93 −0.827396 −0.413698 0.910414i \(-0.635763\pi\)
−0.413698 + 0.910414i \(0.635763\pi\)
\(140\) 2323.86 1.40287
\(141\) 0 0
\(142\) 2215.25 1.30915
\(143\) −1895.31 −1.10835
\(144\) 0 0
\(145\) 1028.29 0.588929
\(146\) −4832.43 −2.73928
\(147\) 0 0
\(148\) −6430.51 −3.57152
\(149\) −259.845 −0.142868 −0.0714340 0.997445i \(-0.522758\pi\)
−0.0714340 + 0.997445i \(0.522758\pi\)
\(150\) 0 0
\(151\) −508.304 −0.273941 −0.136971 0.990575i \(-0.543737\pi\)
−0.136971 + 0.990575i \(0.543737\pi\)
\(152\) −5999.34 −3.20139
\(153\) 0 0
\(154\) −3678.15 −1.92463
\(155\) −91.2485 −0.0472855
\(156\) 0 0
\(157\) −23.3052 −0.0118468 −0.00592342 0.999982i \(-0.501885\pi\)
−0.00592342 + 0.999982i \(0.501885\pi\)
\(158\) −127.050 −0.0639721
\(159\) 0 0
\(160\) −1509.12 −0.745665
\(161\) 3777.49 1.84912
\(162\) 0 0
\(163\) −4032.10 −1.93754 −0.968769 0.247964i \(-0.920238\pi\)
−0.968769 + 0.247964i \(0.920238\pi\)
\(164\) 3732.62 1.77725
\(165\) 0 0
\(166\) −2838.83 −1.32733
\(167\) 671.911 0.311341 0.155671 0.987809i \(-0.450246\pi\)
0.155671 + 0.987809i \(0.450246\pi\)
\(168\) 0 0
\(169\) 2079.10 0.946337
\(170\) 1772.69 0.799761
\(171\) 0 0
\(172\) 6378.49 2.82765
\(173\) −1633.53 −0.717889 −0.358944 0.933359i \(-0.616863\pi\)
−0.358944 + 0.933359i \(0.616863\pi\)
\(174\) 0 0
\(175\) 610.034 0.263510
\(176\) 4243.57 1.81745
\(177\) 0 0
\(178\) −437.549 −0.184245
\(179\) −341.260 −0.142497 −0.0712485 0.997459i \(-0.522698\pi\)
−0.0712485 + 0.997459i \(0.522698\pi\)
\(180\) 0 0
\(181\) −1695.92 −0.696447 −0.348223 0.937412i \(-0.613215\pi\)
−0.348223 + 0.937412i \(0.613215\pi\)
\(182\) 8298.45 3.37979
\(183\) 0 0
\(184\) −8893.90 −3.56341
\(185\) −1688.07 −0.670860
\(186\) 0 0
\(187\) −1975.88 −0.772678
\(188\) 95.3107 0.0369747
\(189\) 0 0
\(190\) −2715.38 −1.03681
\(191\) 726.451 0.275205 0.137603 0.990488i \(-0.456060\pi\)
0.137603 + 0.990488i \(0.456060\pi\)
\(192\) 0 0
\(193\) 4247.26 1.58406 0.792032 0.610479i \(-0.209023\pi\)
0.792032 + 0.610479i \(0.209023\pi\)
\(194\) −4304.14 −1.59288
\(195\) 0 0
\(196\) 4807.94 1.75216
\(197\) −2678.52 −0.968713 −0.484357 0.874871i \(-0.660946\pi\)
−0.484357 + 0.874871i \(0.660946\pi\)
\(198\) 0 0
\(199\) 1486.48 0.529517 0.264759 0.964315i \(-0.414708\pi\)
0.264759 + 0.964315i \(0.414708\pi\)
\(200\) −1436.29 −0.507806
\(201\) 0 0
\(202\) 4283.15 1.49189
\(203\) 5018.32 1.73506
\(204\) 0 0
\(205\) 979.845 0.333831
\(206\) −6090.97 −2.06009
\(207\) 0 0
\(208\) −9574.12 −3.19157
\(209\) 3026.62 1.00170
\(210\) 0 0
\(211\) −4827.41 −1.57504 −0.787519 0.616291i \(-0.788635\pi\)
−0.787519 + 0.616291i \(0.788635\pi\)
\(212\) 6094.37 1.97435
\(213\) 0 0
\(214\) 5321.37 1.69982
\(215\) 1674.41 0.531134
\(216\) 0 0
\(217\) −445.317 −0.139309
\(218\) 2099.23 0.652193
\(219\) 0 0
\(220\) 2760.27 0.845899
\(221\) 4457.88 1.35688
\(222\) 0 0
\(223\) 2774.48 0.833153 0.416576 0.909101i \(-0.363230\pi\)
0.416576 + 0.909101i \(0.363230\pi\)
\(224\) −7364.91 −2.19683
\(225\) 0 0
\(226\) 5628.19 1.65655
\(227\) 5101.34 1.49158 0.745788 0.666184i \(-0.232073\pi\)
0.745788 + 0.666184i \(0.232073\pi\)
\(228\) 0 0
\(229\) −4097.83 −1.18250 −0.591249 0.806489i \(-0.701365\pi\)
−0.591249 + 0.806489i \(0.701365\pi\)
\(230\) −4025.49 −1.15406
\(231\) 0 0
\(232\) −11815.4 −3.34361
\(233\) −357.613 −0.100549 −0.0502747 0.998735i \(-0.516010\pi\)
−0.0502747 + 0.998735i \(0.516010\pi\)
\(234\) 0 0
\(235\) 25.0199 0.00694519
\(236\) −8201.83 −2.26226
\(237\) 0 0
\(238\) 8651.22 2.35620
\(239\) 351.682 0.0951818 0.0475909 0.998867i \(-0.484846\pi\)
0.0475909 + 0.998867i \(0.484846\pi\)
\(240\) 0 0
\(241\) −6165.53 −1.64795 −0.823976 0.566624i \(-0.808249\pi\)
−0.823976 + 0.566624i \(0.808249\pi\)
\(242\) 2553.19 0.678204
\(243\) 0 0
\(244\) 11325.0 2.97134
\(245\) 1262.13 0.329120
\(246\) 0 0
\(247\) −6828.50 −1.75906
\(248\) 1048.48 0.268461
\(249\) 0 0
\(250\) −650.084 −0.164460
\(251\) −3245.53 −0.816160 −0.408080 0.912946i \(-0.633802\pi\)
−0.408080 + 0.912946i \(0.633802\pi\)
\(252\) 0 0
\(253\) 4486.90 1.11498
\(254\) 4026.00 0.994543
\(255\) 0 0
\(256\) −4969.25 −1.21320
\(257\) −3552.19 −0.862178 −0.431089 0.902309i \(-0.641871\pi\)
−0.431089 + 0.902309i \(0.641871\pi\)
\(258\) 0 0
\(259\) −8238.22 −1.97644
\(260\) −6227.59 −1.48546
\(261\) 0 0
\(262\) −6313.81 −1.48881
\(263\) −4416.59 −1.03551 −0.517754 0.855530i \(-0.673232\pi\)
−0.517754 + 0.855530i \(0.673232\pi\)
\(264\) 0 0
\(265\) 1599.83 0.370855
\(266\) −13251.8 −3.05458
\(267\) 0 0
\(268\) 3730.85 0.850366
\(269\) 3419.93 0.775155 0.387578 0.921837i \(-0.373312\pi\)
0.387578 + 0.921837i \(0.373312\pi\)
\(270\) 0 0
\(271\) 716.407 0.160585 0.0802927 0.996771i \(-0.474415\pi\)
0.0802927 + 0.996771i \(0.474415\pi\)
\(272\) −9981.12 −2.22498
\(273\) 0 0
\(274\) 11962.3 2.63748
\(275\) 724.597 0.158890
\(276\) 0 0
\(277\) 657.529 0.142625 0.0713124 0.997454i \(-0.477281\pi\)
0.0713124 + 0.997454i \(0.477281\pi\)
\(278\) 7051.72 1.52135
\(279\) 0 0
\(280\) −7009.49 −1.49606
\(281\) −1513.91 −0.321397 −0.160698 0.987004i \(-0.551375\pi\)
−0.160698 + 0.987004i \(0.551375\pi\)
\(282\) 0 0
\(283\) 3906.38 0.820532 0.410266 0.911966i \(-0.365436\pi\)
0.410266 + 0.911966i \(0.365436\pi\)
\(284\) −8113.16 −1.69517
\(285\) 0 0
\(286\) 9856.89 2.03794
\(287\) 4781.91 0.983509
\(288\) 0 0
\(289\) −265.611 −0.0540629
\(290\) −5347.79 −1.08287
\(291\) 0 0
\(292\) 17698.3 3.54697
\(293\) 8048.76 1.60483 0.802413 0.596770i \(-0.203549\pi\)
0.802413 + 0.596770i \(0.203549\pi\)
\(294\) 0 0
\(295\) −2153.05 −0.424934
\(296\) 19396.4 3.80877
\(297\) 0 0
\(298\) 1351.37 0.262693
\(299\) −10123.1 −1.95797
\(300\) 0 0
\(301\) 8171.57 1.56479
\(302\) 2643.52 0.503700
\(303\) 0 0
\(304\) 15288.9 2.88447
\(305\) 2972.91 0.558125
\(306\) 0 0
\(307\) 101.564 0.0188814 0.00944068 0.999955i \(-0.496995\pi\)
0.00944068 + 0.999955i \(0.496995\pi\)
\(308\) 13470.9 2.49213
\(309\) 0 0
\(310\) 474.553 0.0869445
\(311\) 7684.59 1.40113 0.700567 0.713586i \(-0.252930\pi\)
0.700567 + 0.713586i \(0.252930\pi\)
\(312\) 0 0
\(313\) −1345.15 −0.242915 −0.121457 0.992597i \(-0.538757\pi\)
−0.121457 + 0.992597i \(0.538757\pi\)
\(314\) 121.202 0.0217830
\(315\) 0 0
\(316\) 465.310 0.0828346
\(317\) −7622.33 −1.35051 −0.675257 0.737583i \(-0.735967\pi\)
−0.675257 + 0.737583i \(0.735967\pi\)
\(318\) 0 0
\(319\) 5960.76 1.04620
\(320\) 1991.98 0.347985
\(321\) 0 0
\(322\) −19645.5 −3.40000
\(323\) −7118.78 −1.22631
\(324\) 0 0
\(325\) −1634.80 −0.279022
\(326\) 20969.6 3.56258
\(327\) 0 0
\(328\) −11258.7 −1.89531
\(329\) 122.104 0.0204614
\(330\) 0 0
\(331\) −6585.09 −1.09350 −0.546751 0.837295i \(-0.684136\pi\)
−0.546751 + 0.837295i \(0.684136\pi\)
\(332\) 10397.0 1.71870
\(333\) 0 0
\(334\) −3494.39 −0.572468
\(335\) 979.382 0.159729
\(336\) 0 0
\(337\) −2946.94 −0.476351 −0.238175 0.971222i \(-0.576549\pi\)
−0.238175 + 0.971222i \(0.576549\pi\)
\(338\) −10812.7 −1.74004
\(339\) 0 0
\(340\) −6492.33 −1.03558
\(341\) −528.947 −0.0840002
\(342\) 0 0
\(343\) −2210.14 −0.347919
\(344\) −19239.5 −3.01548
\(345\) 0 0
\(346\) 8495.44 1.31999
\(347\) −8493.48 −1.31399 −0.656994 0.753896i \(-0.728172\pi\)
−0.656994 + 0.753896i \(0.728172\pi\)
\(348\) 0 0
\(349\) 5646.54 0.866053 0.433027 0.901381i \(-0.357446\pi\)
0.433027 + 0.901381i \(0.357446\pi\)
\(350\) −3172.58 −0.484519
\(351\) 0 0
\(352\) −8748.03 −1.32464
\(353\) 1221.93 0.184240 0.0921202 0.995748i \(-0.470636\pi\)
0.0921202 + 0.995748i \(0.470636\pi\)
\(354\) 0 0
\(355\) −2129.78 −0.318414
\(356\) 1602.48 0.238571
\(357\) 0 0
\(358\) 1774.78 0.262011
\(359\) 4151.44 0.610319 0.305160 0.952301i \(-0.401290\pi\)
0.305160 + 0.952301i \(0.401290\pi\)
\(360\) 0 0
\(361\) 4045.41 0.589796
\(362\) 8819.93 1.28057
\(363\) 0 0
\(364\) −30392.3 −4.37635
\(365\) 4645.96 0.666249
\(366\) 0 0
\(367\) −7038.71 −1.00114 −0.500569 0.865696i \(-0.666876\pi\)
−0.500569 + 0.865696i \(0.666876\pi\)
\(368\) 22665.5 3.21065
\(369\) 0 0
\(370\) 8779.08 1.23352
\(371\) 7807.58 1.09259
\(372\) 0 0
\(373\) −7119.57 −0.988303 −0.494152 0.869376i \(-0.664521\pi\)
−0.494152 + 0.869376i \(0.664521\pi\)
\(374\) 10275.9 1.42073
\(375\) 0 0
\(376\) −287.487 −0.0394309
\(377\) −13448.4 −1.83720
\(378\) 0 0
\(379\) 3372.29 0.457053 0.228526 0.973538i \(-0.426609\pi\)
0.228526 + 0.973538i \(0.426609\pi\)
\(380\) 9944.82 1.34252
\(381\) 0 0
\(382\) −3778.03 −0.506024
\(383\) −3958.63 −0.528138 −0.264069 0.964504i \(-0.585065\pi\)
−0.264069 + 0.964504i \(0.585065\pi\)
\(384\) 0 0
\(385\) 3536.23 0.468111
\(386\) −22088.6 −2.91264
\(387\) 0 0
\(388\) 15763.5 2.06256
\(389\) −9654.01 −1.25830 −0.629148 0.777285i \(-0.716596\pi\)
−0.629148 + 0.777285i \(0.716596\pi\)
\(390\) 0 0
\(391\) −10553.4 −1.36499
\(392\) −14502.3 −1.86856
\(393\) 0 0
\(394\) 13930.1 1.78119
\(395\) 122.148 0.0155593
\(396\) 0 0
\(397\) 10928.3 1.38155 0.690776 0.723068i \(-0.257269\pi\)
0.690776 + 0.723068i \(0.257269\pi\)
\(398\) −7730.70 −0.973631
\(399\) 0 0
\(400\) 3660.28 0.457536
\(401\) −4085.57 −0.508787 −0.254393 0.967101i \(-0.581876\pi\)
−0.254393 + 0.967101i \(0.581876\pi\)
\(402\) 0 0
\(403\) 1193.38 0.147510
\(404\) −15686.6 −1.93178
\(405\) 0 0
\(406\) −26098.7 −3.19028
\(407\) −9785.34 −1.19175
\(408\) 0 0
\(409\) 10156.3 1.22786 0.613930 0.789361i \(-0.289588\pi\)
0.613930 + 0.789361i \(0.289588\pi\)
\(410\) −5095.85 −0.613820
\(411\) 0 0
\(412\) 22307.6 2.66752
\(413\) −10507.5 −1.25191
\(414\) 0 0
\(415\) 2729.29 0.322833
\(416\) 19736.9 2.32615
\(417\) 0 0
\(418\) −15740.4 −1.84184
\(419\) −15878.8 −1.85139 −0.925693 0.378275i \(-0.876517\pi\)
−0.925693 + 0.378275i \(0.876517\pi\)
\(420\) 0 0
\(421\) −2279.85 −0.263926 −0.131963 0.991255i \(-0.542128\pi\)
−0.131963 + 0.991255i \(0.542128\pi\)
\(422\) 25105.8 2.89604
\(423\) 0 0
\(424\) −18382.5 −2.10551
\(425\) −1704.29 −0.194519
\(426\) 0 0
\(427\) 14508.6 1.64431
\(428\) −19489.0 −2.20102
\(429\) 0 0
\(430\) −8708.05 −0.976604
\(431\) 1947.38 0.217638 0.108819 0.994062i \(-0.465293\pi\)
0.108819 + 0.994062i \(0.465293\pi\)
\(432\) 0 0
\(433\) 12636.2 1.40244 0.701219 0.712946i \(-0.252639\pi\)
0.701219 + 0.712946i \(0.252639\pi\)
\(434\) 2315.95 0.256150
\(435\) 0 0
\(436\) −7688.25 −0.844496
\(437\) 16165.6 1.76957
\(438\) 0 0
\(439\) −15849.8 −1.72317 −0.861585 0.507614i \(-0.830528\pi\)
−0.861585 + 0.507614i \(0.830528\pi\)
\(440\) −8325.86 −0.902090
\(441\) 0 0
\(442\) −23184.0 −2.49491
\(443\) 17455.6 1.87210 0.936048 0.351872i \(-0.114455\pi\)
0.936048 + 0.351872i \(0.114455\pi\)
\(444\) 0 0
\(445\) 420.666 0.0448123
\(446\) −14429.2 −1.53193
\(447\) 0 0
\(448\) 9721.41 1.02521
\(449\) 16068.1 1.68887 0.844435 0.535658i \(-0.179936\pi\)
0.844435 + 0.535658i \(0.179936\pi\)
\(450\) 0 0
\(451\) 5679.94 0.593033
\(452\) −20612.7 −2.14500
\(453\) 0 0
\(454\) −26530.4 −2.74258
\(455\) −7978.25 −0.822036
\(456\) 0 0
\(457\) 11891.7 1.21722 0.608612 0.793468i \(-0.291727\pi\)
0.608612 + 0.793468i \(0.291727\pi\)
\(458\) 21311.4 2.17428
\(459\) 0 0
\(460\) 14743.0 1.49434
\(461\) 2802.23 0.283108 0.141554 0.989931i \(-0.454790\pi\)
0.141554 + 0.989931i \(0.454790\pi\)
\(462\) 0 0
\(463\) −12933.3 −1.29819 −0.649096 0.760707i \(-0.724853\pi\)
−0.649096 + 0.760707i \(0.724853\pi\)
\(464\) 30110.6 3.01261
\(465\) 0 0
\(466\) 1859.83 0.184882
\(467\) −5748.11 −0.569573 −0.284787 0.958591i \(-0.591923\pi\)
−0.284787 + 0.958591i \(0.591923\pi\)
\(468\) 0 0
\(469\) 4779.65 0.470583
\(470\) −130.120 −0.0127702
\(471\) 0 0
\(472\) 24739.3 2.41254
\(473\) 9706.17 0.943531
\(474\) 0 0
\(475\) 2610.60 0.252174
\(476\) −31684.3 −3.05094
\(477\) 0 0
\(478\) −1828.98 −0.175012
\(479\) 11217.3 1.07000 0.535002 0.844851i \(-0.320311\pi\)
0.535002 + 0.844851i \(0.320311\pi\)
\(480\) 0 0
\(481\) 22077.2 2.09279
\(482\) 32064.9 3.03012
\(483\) 0 0
\(484\) −9350.83 −0.878177
\(485\) 4138.07 0.387423
\(486\) 0 0
\(487\) −8905.12 −0.828603 −0.414301 0.910140i \(-0.635974\pi\)
−0.414301 + 0.910140i \(0.635974\pi\)
\(488\) −34159.7 −3.16872
\(489\) 0 0
\(490\) −6563.91 −0.605157
\(491\) 6553.10 0.602316 0.301158 0.953574i \(-0.402627\pi\)
0.301158 + 0.953574i \(0.402627\pi\)
\(492\) 0 0
\(493\) −14020.1 −1.28079
\(494\) 35512.8 3.23440
\(495\) 0 0
\(496\) −2671.96 −0.241884
\(497\) −10393.9 −0.938087
\(498\) 0 0
\(499\) −4610.09 −0.413579 −0.206789 0.978385i \(-0.566302\pi\)
−0.206789 + 0.978385i \(0.566302\pi\)
\(500\) 2380.87 0.212952
\(501\) 0 0
\(502\) 16878.9 1.50069
\(503\) −13069.1 −1.15850 −0.579249 0.815151i \(-0.696654\pi\)
−0.579249 + 0.815151i \(0.696654\pi\)
\(504\) 0 0
\(505\) −4117.88 −0.362858
\(506\) −23334.9 −2.05012
\(507\) 0 0
\(508\) −14744.9 −1.28779
\(509\) −15930.8 −1.38727 −0.693635 0.720327i \(-0.743992\pi\)
−0.693635 + 0.720327i \(0.743992\pi\)
\(510\) 0 0
\(511\) 22673.6 1.96286
\(512\) 23102.1 1.99410
\(513\) 0 0
\(514\) 18473.8 1.58530
\(515\) 5855.95 0.501056
\(516\) 0 0
\(517\) 145.035 0.0123378
\(518\) 42844.3 3.63411
\(519\) 0 0
\(520\) 18784.4 1.58413
\(521\) −3654.38 −0.307296 −0.153648 0.988126i \(-0.549102\pi\)
−0.153648 + 0.988126i \(0.549102\pi\)
\(522\) 0 0
\(523\) −5138.66 −0.429633 −0.214816 0.976654i \(-0.568915\pi\)
−0.214816 + 0.976654i \(0.568915\pi\)
\(524\) 23123.7 1.92780
\(525\) 0 0
\(526\) 22969.2 1.90400
\(527\) 1244.11 0.102836
\(528\) 0 0
\(529\) 11798.1 0.969681
\(530\) −8320.16 −0.681896
\(531\) 0 0
\(532\) 48533.4 3.95524
\(533\) −12814.8 −1.04141
\(534\) 0 0
\(535\) −5116.04 −0.413431
\(536\) −11253.4 −0.906854
\(537\) 0 0
\(538\) −17785.9 −1.42529
\(539\) 7316.27 0.584664
\(540\) 0 0
\(541\) 6932.06 0.550892 0.275446 0.961317i \(-0.411175\pi\)
0.275446 + 0.961317i \(0.411175\pi\)
\(542\) −3725.80 −0.295271
\(543\) 0 0
\(544\) 20575.9 1.62166
\(545\) −2018.23 −0.158627
\(546\) 0 0
\(547\) −3423.11 −0.267572 −0.133786 0.991010i \(-0.542713\pi\)
−0.133786 + 0.991010i \(0.542713\pi\)
\(548\) −43810.8 −3.41516
\(549\) 0 0
\(550\) −3768.39 −0.292154
\(551\) 21475.6 1.66042
\(552\) 0 0
\(553\) 596.115 0.0458398
\(554\) −3419.59 −0.262246
\(555\) 0 0
\(556\) −25826.3 −1.96992
\(557\) 24489.2 1.86291 0.931455 0.363856i \(-0.118540\pi\)
0.931455 + 0.363856i \(0.118540\pi\)
\(558\) 0 0
\(559\) −21898.6 −1.65691
\(560\) 17863.2 1.34796
\(561\) 0 0
\(562\) 7873.37 0.590957
\(563\) 10053.1 0.752552 0.376276 0.926508i \(-0.377204\pi\)
0.376276 + 0.926508i \(0.377204\pi\)
\(564\) 0 0
\(565\) −5411.02 −0.402908
\(566\) −20315.8 −1.50872
\(567\) 0 0
\(568\) 24471.8 1.80777
\(569\) 6670.45 0.491459 0.245729 0.969338i \(-0.420973\pi\)
0.245729 + 0.969338i \(0.420973\pi\)
\(570\) 0 0
\(571\) 4633.55 0.339594 0.169797 0.985479i \(-0.445689\pi\)
0.169797 + 0.985479i \(0.445689\pi\)
\(572\) −36099.9 −2.63884
\(573\) 0 0
\(574\) −24869.1 −1.80839
\(575\) 3870.17 0.280691
\(576\) 0 0
\(577\) 7045.15 0.508307 0.254154 0.967164i \(-0.418203\pi\)
0.254154 + 0.967164i \(0.418203\pi\)
\(578\) 1381.35 0.0994062
\(579\) 0 0
\(580\) 19585.8 1.40216
\(581\) 13319.7 0.951108
\(582\) 0 0
\(583\) 9273.82 0.658804
\(584\) −53383.7 −3.78259
\(585\) 0 0
\(586\) −41859.0 −2.95082
\(587\) 8001.06 0.562588 0.281294 0.959622i \(-0.409236\pi\)
0.281294 + 0.959622i \(0.409236\pi\)
\(588\) 0 0
\(589\) −1905.71 −0.133316
\(590\) 11197.3 0.781333
\(591\) 0 0
\(592\) −49430.4 −3.43172
\(593\) −6747.53 −0.467265 −0.233632 0.972325i \(-0.575061\pi\)
−0.233632 + 0.972325i \(0.575061\pi\)
\(594\) 0 0
\(595\) −8317.41 −0.573077
\(596\) −4949.26 −0.340150
\(597\) 0 0
\(598\) 52646.9 3.60016
\(599\) −21547.3 −1.46978 −0.734890 0.678186i \(-0.762766\pi\)
−0.734890 + 0.678186i \(0.762766\pi\)
\(600\) 0 0
\(601\) −12155.1 −0.824983 −0.412492 0.910961i \(-0.635341\pi\)
−0.412492 + 0.910961i \(0.635341\pi\)
\(602\) −42497.6 −2.87720
\(603\) 0 0
\(604\) −9681.64 −0.652219
\(605\) −2454.68 −0.164953
\(606\) 0 0
\(607\) 16348.9 1.09322 0.546608 0.837388i \(-0.315919\pi\)
0.546608 + 0.837388i \(0.315919\pi\)
\(608\) −31517.7 −2.10232
\(609\) 0 0
\(610\) −15461.1 −1.02623
\(611\) −327.220 −0.0216660
\(612\) 0 0
\(613\) −29955.5 −1.97372 −0.986859 0.161581i \(-0.948341\pi\)
−0.986859 + 0.161581i \(0.948341\pi\)
\(614\) −528.202 −0.0347174
\(615\) 0 0
\(616\) −40632.4 −2.65767
\(617\) −2159.74 −0.140921 −0.0704603 0.997515i \(-0.522447\pi\)
−0.0704603 + 0.997515i \(0.522447\pi\)
\(618\) 0 0
\(619\) 22100.8 1.43507 0.717535 0.696523i \(-0.245270\pi\)
0.717535 + 0.696523i \(0.245270\pi\)
\(620\) −1738.01 −0.112581
\(621\) 0 0
\(622\) −39965.0 −2.57629
\(623\) 2052.96 0.132023
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) 6995.68 0.446651
\(627\) 0 0
\(628\) −443.893 −0.0282058
\(629\) 23015.7 1.45898
\(630\) 0 0
\(631\) 18360.1 1.15833 0.579164 0.815211i \(-0.303379\pi\)
0.579164 + 0.815211i \(0.303379\pi\)
\(632\) −1403.52 −0.0883372
\(633\) 0 0
\(634\) 39641.2 2.48321
\(635\) −3870.66 −0.241893
\(636\) 0 0
\(637\) −16506.6 −1.02671
\(638\) −30999.9 −1.92367
\(639\) 0 0
\(640\) 1713.32 0.105820
\(641\) 21064.5 1.29797 0.648985 0.760802i \(-0.275194\pi\)
0.648985 + 0.760802i \(0.275194\pi\)
\(642\) 0 0
\(643\) −10539.1 −0.646381 −0.323190 0.946334i \(-0.604755\pi\)
−0.323190 + 0.946334i \(0.604755\pi\)
\(644\) 71949.8 4.40251
\(645\) 0 0
\(646\) 37022.4 2.25484
\(647\) −22553.4 −1.37043 −0.685213 0.728343i \(-0.740291\pi\)
−0.685213 + 0.728343i \(0.740291\pi\)
\(648\) 0 0
\(649\) −12480.8 −0.754873
\(650\) 8502.04 0.513043
\(651\) 0 0
\(652\) −76799.4 −4.61303
\(653\) −22624.0 −1.35582 −0.677908 0.735147i \(-0.737113\pi\)
−0.677908 + 0.735147i \(0.737113\pi\)
\(654\) 0 0
\(655\) 6070.19 0.362110
\(656\) 28692.1 1.70768
\(657\) 0 0
\(658\) −635.022 −0.0376227
\(659\) −6376.60 −0.376930 −0.188465 0.982080i \(-0.560351\pi\)
−0.188465 + 0.982080i \(0.560351\pi\)
\(660\) 0 0
\(661\) −22097.5 −1.30029 −0.650146 0.759809i \(-0.725292\pi\)
−0.650146 + 0.759809i \(0.725292\pi\)
\(662\) 34246.9 2.01064
\(663\) 0 0
\(664\) −31360.5 −1.83287
\(665\) 12740.4 0.742938
\(666\) 0 0
\(667\) 31837.2 1.84819
\(668\) 12797.9 0.741264
\(669\) 0 0
\(670\) −5093.44 −0.293697
\(671\) 17233.3 0.991479
\(672\) 0 0
\(673\) 24033.0 1.37653 0.688264 0.725461i \(-0.258373\pi\)
0.688264 + 0.725461i \(0.258373\pi\)
\(674\) 15326.1 0.875873
\(675\) 0 0
\(676\) 39600.6 2.25311
\(677\) −179.638 −0.0101980 −0.00509901 0.999987i \(-0.501623\pi\)
−0.00509901 + 0.999987i \(0.501623\pi\)
\(678\) 0 0
\(679\) 20194.9 1.14140
\(680\) 19582.9 1.10437
\(681\) 0 0
\(682\) 2750.88 0.154452
\(683\) 30434.2 1.70502 0.852511 0.522709i \(-0.175079\pi\)
0.852511 + 0.522709i \(0.175079\pi\)
\(684\) 0 0
\(685\) −11500.7 −0.641490
\(686\) 11494.2 0.639725
\(687\) 0 0
\(688\) 49030.5 2.71696
\(689\) −20923.1 −1.15691
\(690\) 0 0
\(691\) 9792.73 0.539122 0.269561 0.962983i \(-0.413121\pi\)
0.269561 + 0.962983i \(0.413121\pi\)
\(692\) −31113.7 −1.70920
\(693\) 0 0
\(694\) 44171.8 2.41605
\(695\) −6779.63 −0.370023
\(696\) 0 0
\(697\) −13359.6 −0.726010
\(698\) −29365.8 −1.59242
\(699\) 0 0
\(700\) 11619.3 0.627383
\(701\) −8130.47 −0.438065 −0.219032 0.975718i \(-0.570290\pi\)
−0.219032 + 0.975718i \(0.570290\pi\)
\(702\) 0 0
\(703\) −35255.0 −1.89142
\(704\) 11547.1 0.618177
\(705\) 0 0
\(706\) −6354.86 −0.338765
\(707\) −20096.4 −1.06903
\(708\) 0 0
\(709\) −4859.95 −0.257432 −0.128716 0.991681i \(-0.541086\pi\)
−0.128716 + 0.991681i \(0.541086\pi\)
\(710\) 11076.3 0.585472
\(711\) 0 0
\(712\) −4833.59 −0.254419
\(713\) −2825.17 −0.148392
\(714\) 0 0
\(715\) −9476.55 −0.495669
\(716\) −6499.96 −0.339267
\(717\) 0 0
\(718\) −21590.3 −1.12220
\(719\) −19463.0 −1.00952 −0.504762 0.863259i \(-0.668420\pi\)
−0.504762 + 0.863259i \(0.668420\pi\)
\(720\) 0 0
\(721\) 28578.6 1.47618
\(722\) −21038.8 −1.08447
\(723\) 0 0
\(724\) −32302.2 −1.65815
\(725\) 5141.44 0.263377
\(726\) 0 0
\(727\) 2432.66 0.124102 0.0620512 0.998073i \(-0.480236\pi\)
0.0620512 + 0.998073i \(0.480236\pi\)
\(728\) 91672.8 4.66706
\(729\) 0 0
\(730\) −24162.1 −1.22504
\(731\) −22829.5 −1.15510
\(732\) 0 0
\(733\) −17967.6 −0.905386 −0.452693 0.891666i \(-0.649537\pi\)
−0.452693 + 0.891666i \(0.649537\pi\)
\(734\) 36606.0 1.84081
\(735\) 0 0
\(736\) −46724.4 −2.34006
\(737\) 5677.26 0.283751
\(738\) 0 0
\(739\) 23473.0 1.16843 0.584214 0.811599i \(-0.301403\pi\)
0.584214 + 0.811599i \(0.301403\pi\)
\(740\) −32152.6 −1.59723
\(741\) 0 0
\(742\) −40604.6 −2.00895
\(743\) 33559.2 1.65702 0.828512 0.559971i \(-0.189188\pi\)
0.828512 + 0.559971i \(0.189188\pi\)
\(744\) 0 0
\(745\) −1299.23 −0.0638925
\(746\) 37026.5 1.81721
\(747\) 0 0
\(748\) −37634.6 −1.83965
\(749\) −24967.6 −1.21802
\(750\) 0 0
\(751\) −7782.75 −0.378158 −0.189079 0.981962i \(-0.560550\pi\)
−0.189079 + 0.981962i \(0.560550\pi\)
\(752\) 732.640 0.0355274
\(753\) 0 0
\(754\) 69940.5 3.37809
\(755\) −2541.52 −0.122510
\(756\) 0 0
\(757\) −38154.3 −1.83189 −0.915946 0.401300i \(-0.868558\pi\)
−0.915946 + 0.401300i \(0.868558\pi\)
\(758\) −17538.2 −0.840390
\(759\) 0 0
\(760\) −29996.7 −1.43170
\(761\) −19867.1 −0.946363 −0.473182 0.880965i \(-0.656895\pi\)
−0.473182 + 0.880965i \(0.656895\pi\)
\(762\) 0 0
\(763\) −9849.52 −0.467335
\(764\) 13836.7 0.655228
\(765\) 0 0
\(766\) 20587.5 0.971094
\(767\) 28158.5 1.32561
\(768\) 0 0
\(769\) −15710.8 −0.736730 −0.368365 0.929681i \(-0.620082\pi\)
−0.368365 + 0.929681i \(0.620082\pi\)
\(770\) −18390.7 −0.860723
\(771\) 0 0
\(772\) 80897.5 3.77146
\(773\) −25811.9 −1.20102 −0.600510 0.799617i \(-0.705036\pi\)
−0.600510 + 0.799617i \(0.705036\pi\)
\(774\) 0 0
\(775\) −456.242 −0.0211467
\(776\) −47547.8 −2.19957
\(777\) 0 0
\(778\) 50207.3 2.31365
\(779\) 20463.9 0.941201
\(780\) 0 0
\(781\) −12345.8 −0.565645
\(782\) 54885.0 2.50982
\(783\) 0 0
\(784\) 36958.0 1.68358
\(785\) −116.526 −0.00529807
\(786\) 0 0
\(787\) −29242.6 −1.32450 −0.662252 0.749281i \(-0.730399\pi\)
−0.662252 + 0.749281i \(0.730399\pi\)
\(788\) −51017.7 −2.30638
\(789\) 0 0
\(790\) −635.252 −0.0286092
\(791\) −26407.2 −1.18702
\(792\) 0 0
\(793\) −38880.8 −1.74111
\(794\) −56834.6 −2.54028
\(795\) 0 0
\(796\) 28313.0 1.26071
\(797\) −32573.7 −1.44771 −0.723853 0.689955i \(-0.757630\pi\)
−0.723853 + 0.689955i \(0.757630\pi\)
\(798\) 0 0
\(799\) −341.130 −0.0151043
\(800\) −7545.60 −0.333472
\(801\) 0 0
\(802\) 21247.7 0.935514
\(803\) 26931.6 1.18356
\(804\) 0 0
\(805\) 18887.5 0.826951
\(806\) −6206.39 −0.271229
\(807\) 0 0
\(808\) 47315.8 2.06010
\(809\) 34644.9 1.50562 0.752812 0.658236i \(-0.228697\pi\)
0.752812 + 0.658236i \(0.228697\pi\)
\(810\) 0 0
\(811\) −29057.9 −1.25815 −0.629077 0.777343i \(-0.716567\pi\)
−0.629077 + 0.777343i \(0.716567\pi\)
\(812\) 95583.9 4.13096
\(813\) 0 0
\(814\) 50890.3 2.19128
\(815\) −20160.5 −0.866493
\(816\) 0 0
\(817\) 34969.8 1.49748
\(818\) −52819.3 −2.25768
\(819\) 0 0
\(820\) 18663.1 0.794809
\(821\) −46709.5 −1.98560 −0.992798 0.119802i \(-0.961774\pi\)
−0.992798 + 0.119802i \(0.961774\pi\)
\(822\) 0 0
\(823\) −3468.10 −0.146890 −0.0734450 0.997299i \(-0.523399\pi\)
−0.0734450 + 0.997299i \(0.523399\pi\)
\(824\) −67286.8 −2.84472
\(825\) 0 0
\(826\) 54645.9 2.30191
\(827\) −42454.9 −1.78513 −0.892564 0.450920i \(-0.851096\pi\)
−0.892564 + 0.450920i \(0.851096\pi\)
\(828\) 0 0
\(829\) −3933.47 −0.164795 −0.0823975 0.996600i \(-0.526258\pi\)
−0.0823975 + 0.996600i \(0.526258\pi\)
\(830\) −14194.2 −0.593598
\(831\) 0 0
\(832\) −26051.9 −1.08556
\(833\) −17208.3 −0.715764
\(834\) 0 0
\(835\) 3359.55 0.139236
\(836\) 57647.9 2.38492
\(837\) 0 0
\(838\) 82580.5 3.40417
\(839\) 32959.6 1.35625 0.678123 0.734948i \(-0.262794\pi\)
0.678123 + 0.734948i \(0.262794\pi\)
\(840\) 0 0
\(841\) 17906.1 0.734188
\(842\) 11856.7 0.485285
\(843\) 0 0
\(844\) −91947.6 −3.74996
\(845\) 10395.5 0.423215
\(846\) 0 0
\(847\) −11979.5 −0.485974
\(848\) 46846.5 1.89707
\(849\) 0 0
\(850\) 8863.47 0.357664
\(851\) −52264.8 −2.10530
\(852\) 0 0
\(853\) 38845.8 1.55927 0.779634 0.626235i \(-0.215405\pi\)
0.779634 + 0.626235i \(0.215405\pi\)
\(854\) −75454.3 −3.02341
\(855\) 0 0
\(856\) 58785.0 2.34723
\(857\) 29305.3 1.16809 0.584043 0.811723i \(-0.301470\pi\)
0.584043 + 0.811723i \(0.301470\pi\)
\(858\) 0 0
\(859\) −909.659 −0.0361318 −0.0180659 0.999837i \(-0.505751\pi\)
−0.0180659 + 0.999837i \(0.505751\pi\)
\(860\) 31892.4 1.26456
\(861\) 0 0
\(862\) −10127.7 −0.400173
\(863\) −47998.0 −1.89324 −0.946622 0.322346i \(-0.895529\pi\)
−0.946622 + 0.322346i \(0.895529\pi\)
\(864\) 0 0
\(865\) −8167.64 −0.321050
\(866\) −65716.6 −2.57868
\(867\) 0 0
\(868\) −8481.94 −0.331677
\(869\) 708.065 0.0276403
\(870\) 0 0
\(871\) −12808.7 −0.498286
\(872\) 23190.2 0.900594
\(873\) 0 0
\(874\) −84071.7 −3.25374
\(875\) 3050.17 0.117845
\(876\) 0 0
\(877\) 3258.01 0.125445 0.0627225 0.998031i \(-0.480022\pi\)
0.0627225 + 0.998031i \(0.480022\pi\)
\(878\) 82429.8 3.16842
\(879\) 0 0
\(880\) 21217.8 0.812788
\(881\) 33380.2 1.27651 0.638256 0.769824i \(-0.279656\pi\)
0.638256 + 0.769824i \(0.279656\pi\)
\(882\) 0 0
\(883\) 33714.6 1.28492 0.642460 0.766319i \(-0.277914\pi\)
0.642460 + 0.766319i \(0.277914\pi\)
\(884\) 84909.2 3.23055
\(885\) 0 0
\(886\) −90780.6 −3.44225
\(887\) 6218.80 0.235408 0.117704 0.993049i \(-0.462447\pi\)
0.117704 + 0.993049i \(0.462447\pi\)
\(888\) 0 0
\(889\) −18889.8 −0.712649
\(890\) −2187.75 −0.0823971
\(891\) 0 0
\(892\) 52845.5 1.98363
\(893\) 522.537 0.0195812
\(894\) 0 0
\(895\) −1706.30 −0.0637266
\(896\) 8361.46 0.311760
\(897\) 0 0
\(898\) −83565.1 −3.10535
\(899\) −3753.19 −0.139239
\(900\) 0 0
\(901\) −21812.6 −0.806529
\(902\) −29539.5 −1.09042
\(903\) 0 0
\(904\) 62174.4 2.28749
\(905\) −8479.61 −0.311460
\(906\) 0 0
\(907\) −22878.6 −0.837566 −0.418783 0.908086i \(-0.637543\pi\)
−0.418783 + 0.908086i \(0.637543\pi\)
\(908\) 97165.0 3.55125
\(909\) 0 0
\(910\) 41492.3 1.51149
\(911\) 30144.8 1.09631 0.548157 0.836376i \(-0.315330\pi\)
0.548157 + 0.836376i \(0.315330\pi\)
\(912\) 0 0
\(913\) 15821.1 0.573496
\(914\) −61844.9 −2.23813
\(915\) 0 0
\(916\) −78051.2 −2.81538
\(917\) 29624.1 1.06682
\(918\) 0 0
\(919\) 3803.52 0.136525 0.0682625 0.997667i \(-0.478254\pi\)
0.0682625 + 0.997667i \(0.478254\pi\)
\(920\) −44469.5 −1.59360
\(921\) 0 0
\(922\) −14573.5 −0.520555
\(923\) 27854.0 0.993312
\(924\) 0 0
\(925\) −8440.33 −0.300018
\(926\) 67262.0 2.38700
\(927\) 0 0
\(928\) −62072.5 −2.19572
\(929\) −125.985 −0.00444934 −0.00222467 0.999998i \(-0.500708\pi\)
−0.00222467 + 0.999998i \(0.500708\pi\)
\(930\) 0 0
\(931\) 26359.3 0.927918
\(932\) −6811.45 −0.239395
\(933\) 0 0
\(934\) 29894.0 1.04728
\(935\) −9879.41 −0.345552
\(936\) 0 0
\(937\) 28107.9 0.979984 0.489992 0.871727i \(-0.337000\pi\)
0.489992 + 0.871727i \(0.337000\pi\)
\(938\) −24857.4 −0.865268
\(939\) 0 0
\(940\) 476.554 0.0165356
\(941\) 49194.9 1.70426 0.852130 0.523330i \(-0.175311\pi\)
0.852130 + 0.523330i \(0.175311\pi\)
\(942\) 0 0
\(943\) 30337.3 1.04763
\(944\) −63046.3 −2.17371
\(945\) 0 0
\(946\) −50478.6 −1.73488
\(947\) 14498.0 0.497490 0.248745 0.968569i \(-0.419982\pi\)
0.248745 + 0.968569i \(0.419982\pi\)
\(948\) 0 0
\(949\) −60761.7 −2.07841
\(950\) −13576.9 −0.463676
\(951\) 0 0
\(952\) 95569.8 3.25361
\(953\) −3201.79 −0.108831 −0.0544155 0.998518i \(-0.517330\pi\)
−0.0544155 + 0.998518i \(0.517330\pi\)
\(954\) 0 0
\(955\) 3632.26 0.123075
\(956\) 6698.48 0.226616
\(957\) 0 0
\(958\) −58337.6 −1.96743
\(959\) −56126.7 −1.88991
\(960\) 0 0
\(961\) −29457.9 −0.988820
\(962\) −114816. −3.84805
\(963\) 0 0
\(964\) −117435. −3.92356
\(965\) 21236.3 0.708415
\(966\) 0 0
\(967\) 57781.9 1.92155 0.960776 0.277326i \(-0.0894481\pi\)
0.960776 + 0.277326i \(0.0894481\pi\)
\(968\) 28205.1 0.936513
\(969\) 0 0
\(970\) −21520.7 −0.712359
\(971\) −2611.60 −0.0863133 −0.0431566 0.999068i \(-0.513741\pi\)
−0.0431566 + 0.999068i \(0.513741\pi\)
\(972\) 0 0
\(973\) −33086.4 −1.09013
\(974\) 46312.6 1.52356
\(975\) 0 0
\(976\) 87053.4 2.85503
\(977\) 33598.3 1.10021 0.550104 0.835096i \(-0.314588\pi\)
0.550104 + 0.835096i \(0.314588\pi\)
\(978\) 0 0
\(979\) 2438.51 0.0796067
\(980\) 24039.7 0.783592
\(981\) 0 0
\(982\) −34080.5 −1.10749
\(983\) 39484.8 1.28115 0.640575 0.767895i \(-0.278696\pi\)
0.640575 + 0.767895i \(0.278696\pi\)
\(984\) 0 0
\(985\) −13392.6 −0.433222
\(986\) 72913.7 2.35501
\(987\) 0 0
\(988\) −130062. −4.18809
\(989\) 51841.9 1.66681
\(990\) 0 0
\(991\) 39918.6 1.27957 0.639786 0.768553i \(-0.279023\pi\)
0.639786 + 0.768553i \(0.279023\pi\)
\(992\) 5508.20 0.176296
\(993\) 0 0
\(994\) 54055.2 1.72487
\(995\) 7432.41 0.236807
\(996\) 0 0
\(997\) −25670.3 −0.815432 −0.407716 0.913109i \(-0.633675\pi\)
−0.407716 + 0.913109i \(0.633675\pi\)
\(998\) 23975.6 0.760454
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 135.4.a.f.1.1 3
3.2 odd 2 135.4.a.g.1.3 yes 3
4.3 odd 2 2160.4.a.bm.1.1 3
5.2 odd 4 675.4.b.l.649.1 6
5.3 odd 4 675.4.b.l.649.6 6
5.4 even 2 675.4.a.r.1.3 3
9.2 odd 6 405.4.e.r.271.1 6
9.4 even 3 405.4.e.t.136.3 6
9.5 odd 6 405.4.e.r.136.1 6
9.7 even 3 405.4.e.t.271.3 6
12.11 even 2 2160.4.a.be.1.1 3
15.2 even 4 675.4.b.k.649.6 6
15.8 even 4 675.4.b.k.649.1 6
15.14 odd 2 675.4.a.q.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
135.4.a.f.1.1 3 1.1 even 1 trivial
135.4.a.g.1.3 yes 3 3.2 odd 2
405.4.e.r.136.1 6 9.5 odd 6
405.4.e.r.271.1 6 9.2 odd 6
405.4.e.t.136.3 6 9.4 even 3
405.4.e.t.271.3 6 9.7 even 3
675.4.a.q.1.1 3 15.14 odd 2
675.4.a.r.1.3 3 5.4 even 2
675.4.b.k.649.1 6 15.8 even 4
675.4.b.k.649.6 6 15.2 even 4
675.4.b.l.649.1 6 5.2 odd 4
675.4.b.l.649.6 6 5.3 odd 4
2160.4.a.be.1.1 3 12.11 even 2
2160.4.a.bm.1.1 3 4.3 odd 2